General Topology

If $X$ is irreducible, then $F$ is irreducible.

$F$ is irreducible if the following statement holds: "If $F=F_1\cup F_2$ for two closed subsets $F_1,F_2$ of $F$, then either $F=F_1$ or $F=F_2$."

$F$ is irreducible if the following statement holds: "If $F\subseteq X_1\cup X_2$ for two closed subsets $X_1,X_2$ of $X$, then either $F\subseteq X_1$ or $F\subseteq X_2$."

If the closure $\overline{F}$ (of $F$ in $X$) is irreducible, then so is $F$.

If $F$ is the closure $\overline{S}$ of an irreducible subspace $S$ of $X$, then $F$ itself is irreducible.